1. Field of the Invention
The invention relates generally to the field of estimating material properties of porous media. More specifically, the invention relates to methods for estimating such properties using computer tomographic (CT) images of porous media such as subsurface rock formation.
2. Background Art
Estimating material properties such as effective elastic and shear moduli, electrical resistivity and fluid transport properties of porous media, for example, mobility of hydrocarbon in subsurface rock formations, has substantial economic significance. Methods known in the art for identifying the existence of subsurface hydrocarbon reservoirs, including seismic surveying and well log analysis, need to be supplemented with reliable methods for estimating how fluids disposed in the pore spaces of reservoir rock formations will flow over time in order to characterize the economic value of such reservoir rock formations.
One method known in the art for estimating fluid transport properties is described in U.S. Pat. No. 6,516,080 issued to Nur. The method described in the Nur patent includes preparing a “thin section” from a specimen of rock formation. The preparation typically includes filling the pore spaces with a dyed epoxy resin. A color micrograph of the section is digitized and converted to an n-ary index image, for example a binary index image. Statistical functions are derived from the two-dimensional image and such functions are used to generate three-dimensional representations of the rock formation. Boundaries can be unconditional or conditioned to the two-dimensional n-ary index image. Desired physical property values are estimated by performing numerical simulations on the three-dimensional representations. For example, permeability is estimated by using a Lattice-Boltzmann flow simulation. Typically, multiple, equiprobable three-dimensional representations are generated for each n-ary index image, and the multiple estimated physical property values are averaged to provide a result.
In performing the method described in the Nur patent, it is necessary to obtain samples of the rock formation and to prepare, as explained above, a section to digitize as a color image. Economic considerations make it desirable to obtain input to fluid transport analysis more quickly than can be obtained using prepared sections. Recently, devices for generating CT images of samples such as drill cuttings have become available. Such CT image generating devices (CT scanners) typically produce three-dimensional gray scale images of the samples analyzed in the scanner. Such gray scale images can be used, for example, essentially contemporaneously as drill cuttings are generated during the drilling of a wellbore through subsurface rock formations.
Using images of samples of rock formations, it is possible to obtain estimates of petrophysical parameters of the imaged rock sample, for example, porosity, permeability, shear and bulk moduli, and formation resistivity factor. The foregoing parameters are typically distributed within ranges in each rock formation, and there may be determinable relationships between such parameters such that determining one parameter value can enable determining one or more of the other parameters. One way to establish such relationship is to determine one or more rock physics transforms. A rock physics transform is a mathematical formula that relates one property of a rock formation to another. Such transforms can be based on an idealized mathematical model of rock, such as the differential effective medium that models rock as a solid with ideal-shape inclusions or the Hertz-Mindlin model that models rock as a composite made of perfect elastic spheres. Such transforms can also be based on a sufficient number of experimental data (e.g., well log measurements or laboratory measurements) using a statistically fit expression that approximates such data. An example of the latter is the Raymer transform between porosity φ and the compressional wave (P-wave) velocity of the rock (Vp). The transform is the expression Vp=(1−φ)2Vps+φVpf, where Vps is the P-wave velocity in the mineral (matrix or solid) phase of the rock (e.g., quartz) and Vpf is the P-wave velocity in the pore fluid (e.g., water). The elastic (compressional) wave velocity is directly related to the bulk K and shear G moduli by the expression Vp=√{square root over ((K+4G/3)/ρ)}, where ρ is the bulk density of the rock. The foregoing moduli can be obtained by laboratory measurement, and can also be obtained by calculations made from an image of a rock sample. Another example is the relationship between the absolute permeability k and the porosity φ of a rock formation called the Kozeny-Carman relation, represented by the expression k=d2φ3/[72τ2(1−φ)2], where d is the mean rock grain size and τ is the pore tortuosity (typically represented by a number between 1 and 5). Yet another example is Humble's relationship between the electrical resistivity formation factor F and the porosity φ, represented by the expression F=a/φm, where a and m are constants that are determined experimentally. As in the P-wave velocity example, the parameters that enter these two equations, one for permeability and the other for the formation factor, can be obtained by laboratory measurement and also by calculations based on an image of a rock sample. Instead of using the permeability (k) and formation factor (F) equation examples above, one may conduct a large number of laboratory tests on samples that represent the formation under examination. Alternatively, such data can be obtained by digital calculations on a digitally imaged rock sample.
Obtaining and calibrating permeability measurements using physical samples and using measurements made on actual rock samples require extensive laboratory and/or well measurements. There exists a need to use images such as the foregoing CT scan images to determine permeability characteristics without the need for extensive laboratory or well measurements.